QCE Mathematical Methods: Exam practice Q13 (2024, Paper 2)

Q13 · 2024 · Technology-activeSimple familiar8 marks

QUESTION 13 (8 marks)

The number of termites in a particular nest can be modelled by

N(t)=\dfrac{A}{2+e^{-t}}

, where

A

is a constant and

t

represents time (months) since the nest first became a visible mound above ground level.

It is estimated that when the mound first became visible, the population was

3\times10^5

termites.

Determine the value of

A

[1 mark]

Determine the number of termites in the nest half a year after the mound became visible.

[2 marks]

Determine the time in months after the mound became visible for the initial population to increase by 130 000 termites. Express the time as a decimal.

[2 marks]

Develop a formula for the rate of change in the number of termites at any time after the mound became visible. Express your formula as a fraction.

[2 marks]

Determine the rate of change in the number of termites five months after the mound became visible.

[1 mark]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

8 marks

ANSWER

A=9\times10^5

; b)

449443

termites; c)

2.3749

months; d)

\dfrac{dN}{dt}=\dfrac{9\times10^5}{e^t(2+e^{-t})^2}

; e) approximately

1506

termites/month.

Worked solution

N(0)=\frac{A}{2+e^0}=\frac A3=3\times10^5,

A=9\times10^5.

b) Half a year is

t=6

N(6)=\frac{9\times10^5}{2+e^{-6}}=449442.9711\approx449443\text{ termites}.

c) The required population is

430000

. Solving

430000=\frac{9\times10^5}{2+e^{-t}}

gives

t=2.3749\text{ months}.

N(t)=(9\times10^5)(2+e^{-t})^{-1},

so by the chain rule

\frac{dN}{dt}=-(9\times10^5)(2+e^{-t})^{-2}(-e^{-t})=\frac{9\times10^5}{e^t(2+e^{-t})^2}.

e) Using this formula,

N'(5)=1505.8744\approx1506\text{ termites/month}.

correctly determines the value of A

[1 mark]

correctly determines

t=6

[1 mark]

determines the number of termites

[1 mark]

correctly determines the required number of termites

[1 mark]

determines the time required

[1 mark]

shows application of the chain rule

[1 mark]

determines a formula for the number of termites expressed as a fraction

[1 mark]

determines the rate of change

[1 mark]

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Mathematical Methods — Question 13

QUESTION 13 (8 marks)

QCAA guide · typeset solution

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